# A priori and a posteriori within an axiomatic framework?

I am not a philosopher, so please excuse my limited understanding of these basic philosophy concepts. I am having some difficulty understanding what truly constitutes an a priori statement/truth (I'm not sure which term is more applicable here), particularly whether it can be framework dependent or not. By 'framework' here I mean some set of axioms. I have been considering specific examples:

• There are analytic statements, such as 'all brothers are male', which I think are true a priori, but these rely on there being a definition of 'brother' which includes their being male. I'm not sure if having a 'definition' of a work is equivalent to having an axiom. If this is the case, and this is true a priori, then it seems that a priori truths can be framework dependent.

• On the other hand I don't think that all a priori truths are definition dependent. 'I think, therefore I am' seems to be a non-analytic, a priori truth? Although maybe this is actually exactly the same case as above and it is analytic, based on a definition of 'thinking' as there being someone that is doing it. But if you think about it conceptually rather than with language, I think this could be an a priori truth that is not definition dependent, and not analytic.

• In the above I have only discussed 'definitions' and I am not sure whether or not these count as an 'axiomatic framework'. In general, I am still unsure as to whether truths that are reached deductively within some framework, whether in mathematics or in physics etc, can be considered a priori. For example, the truths about the internal angles of regular polygons that are deduced within the framework of Euclidean geometry. Technically, these facts can be deduced without constructing these polygons and measuring all of the angles. Although, thinking about it now, the method of 'deducing' these facts seems to involve the mental construction of a polygon, splitting it up into triangles etc... So it almost seems a posteriori, but done mentally.

I would greatly appreciate if someone could clear up the relationship/distinctions between a priori/a posteriori, and truths deduced within an axiomatic framework that are framework-dependent. It would also be great if someone could comment on whether definitions of words, which we use in determining analytic truths, count as axioms in a similar way to axioms for some mathematical or scientific framework.

As an aside, these questions stemmed from considering whether there can be any scientific truths that are deduced a priori. I suppose not, because even if we make a valid deduction from some postulates such as in quantum mechanics, the deduction may not correspond with physical reality (which is what I would define to be a scientific truth). There is no reason whatsoever that some scientific framework, the postulates of which are generally deduced inductively from our observations, i.e. a posteriori, is actually 'true' in the scientific sense. In fact, considering this now it seems that the notion of truth itself depends on the context in which we are discussing it. Scientific truth seems intrinsically a posteriori (although the answer in this post seems to suggest that the distinction/overlap if not so clear) while 'mathematical truths', and analytic truths from statements such as in the first example, by definition seem to be context-dependent (i.e. require axioms/definitons). We still seem to regard them as truths however...

NOTE: I have also read this post, in which priori knowledge is defined as 'knowable without any independent experience' in the top answer. However this does not clear up my difficulty!

• I consider a priori and a posteriori as standing in a logical relation to experience: That which is a priori is logically necessary for experience to be the way it is. In the case of analytic statements/definitions for it being an analytic statement or definition, it has to be true. That which is a posteriori true is true because it follows or is content of that which is experienced. Commented Mar 19, 2018 at 17:20
• @PhilipKlöcking Thank you for your comment. If I understand correctly, you would regard a set of observations of how the physical world is as true a posteriori, but any deductions made from this are a priori. Where would induction, such as that used to propose the axioms of quantum mechanics, fit with this? I suppose propositions derived inductively are not truths, so we would not classify them as a priori or a posteriori?
– Meep
Commented Mar 19, 2018 at 17:37
• See A Priori Justification and Knowledge for an overview: "A priori justification is a type of epistemic justification that is, in some sense, independent of experience. For those that do, a priori knowledge is knowledge based on a priori justification" And see examples. Commented Mar 19, 2018 at 18:02